Neka je w ∈ C takav da je w
2 + 10 = w. Definirajmo skup
Ω = {a + bw : a, b ∈ R} .
Je li Ω potpolje od C? Ako da, jesu li Ω i R medusobno izomorfna polja?

### Problem Overview

Given \( w \in \mathbb{C} \) such that \( w^2 + 10 = w \), we define the set \( \Omega = \{a + bw : a, b \in \mathbb{R}\} \). We need to determine if \( \Omega \) is a subfield of \( \mathbb{C} \) and, if so, whether \( \Omega \) and \( \mathbb{R} \) are isomorphic fields.

### Step 1: Simplify the equation \( w^2 + 10 = w \)

We start by rearranging the equation:

\[
w^2 - w + 10 = 0
\]

This is a quadratic equation. To solve for \( w \), we use the quadratic formula:

\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 1 \), \( b = -1 \), and \( c = 10 \):

\[
w = \frac{1 \pm \sqrt{1^2 - 4(1)(10)}}{2(1)} = \frac{1 \pm \sqrt{1 - 40}}{2} = \frac{1 \pm \sqrt{-39}}{2} = \frac{1 \pm i\sqrt{39}}{2}
\]

So, \( w = \frac{1 + i\sqrt{39}}{2} \) or \( w = \frac{1 - i\sqrt{39}}{2} \).

### Step 2: Determine if \( \Omega \) is a subfield of \( \mathbb{C} \)

For \( \Omega \) to be a subfield of \( \mathbb{C} \), it must be closed under addition, subtraction, multiplication, and division (except by 0).

- **Addition and Subtraction:** If \( z_1 = a + bw \) and \( z_2 = c + dw \) are in \( \Omega \), then \( z_1 + z_2 = (a + c) + (b + d)w \) and \( z_1 - z_2 = (a - c) + (b - d)w \) are also in \( \Omega \). Thus, \( \Omega \) is closed under addition and subtraction.

- **Multiplication:** For \( z_1 = a + bw \) and \( z_2 = c + dw \), their product is:

\[
z_1 z_2 = (a + bw)(c + dw) = ac + adw + bcw + bdw^2
\]

Since \( w^2 = w - 10 \), we have:

\[
z_1 z_2 = ac + adw + bcw + bd(w - 10) = (ac - 10bd) + (ad + bc)w
\]

This shows that \( z_1z_2 \in \Omega \). So, \( \Omega \) is closed under multiplication.

- **Division:** For \( z_1 = a + bw \), its inverse \( z_1^{-1} \) must also be in \( \Omega \) (assuming \( z_1 \neq 0 \)). We must show:

\[
z_1^{-1} = \frac{1}{a + bw} = \frac{a - bw}{(a + bw)(a - bw)} = \frac{a - bw}{a^2 - b^2w^2}
\]

Using \( w^2 = w - 10 \), we find \( z_1^{-1} \) can also be expressed as a linear combination of 1 and \( w \), which belongs to \( \Omega \).

Therefore, \( \Omega \) is closed under division.

### Conclusion: \( \Omega \) is a subfield of \( \mathbb{C} \).

### Step 3: Check if \( \Omega \) and \( \mathbb{R} \) are isomorphic

\( \Omega \) is a subfield of \( \mathbb{C} \) containing all elements of the form \( a + bw \), where \( a, b \in \mathbb{R} \). This structure is a degree 2 extension of \( \mathbb{R} \) (since \( w \) is not real), which means \( \Omega \) has dimension 2 over \( \mathbb{R} \).

However, \( \mathbb{R} \) as a field is not isomorphic to any proper extension of itself (like \( \Omega \)). Thus, \( \Omega \) and \( \mathbb{R} \) are **not** isomorphic fields.

### Summary
- **\( \Omega \) is a subfield of \( \mathbb{C} \).**
- **\( \Omega \) and \( \mathbb{R} \) are not isomorphic fields.**

Would you like further details on any part of the solution?

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**Related Questions:**
1. What is the characteristic of the field \( \mathbb{C} \)?
2. How do you determine if two fields are isomorphic?
3. What are the properties of quadratic field extensions?
4. Can \( w \) be expressed in terms of elementary functions?
5. What is the relationship between subfields and vector spaces?

**Tip:** When working with field extensions, always check the degree of the extension to understand the structure and possible isomorphisms.

Explore the concept of subfields in complex field theory by solving the problem involving a complex number w such that w^2 + 10 = w. Discover if the set Omega, defined as the set of all expressions a + bw where a, b are real numbers, forms a subfield of the complex numbers and whether it is isomorphic to the field of real numbers.

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